Clifford Algebra in Euclidean 3-Space

Chapter 2Clifford Algebra in Euclidean 3-Space

2.1 Reflections, Rotations, and Quaternions in E3

2.1.1 Using Square Matrices to Represent Vectors

One frequently represents a vector x in the 3-dimensional Euclidean space E3 byx D xi C yj C zk or .x; y; z/. However, neither of these notations easily generalizeto higher dimensions. Alternate notations which do easily generalizes to higherdimensions are x D x1i1 C x2i2 C x3i3 and

x D �x1; x2; x3

� D x1 .1; 0; 0/C x2 .0; 1; 0/C x3 .0; 0; 1/ : (2.1)

These alternate notations have their own problem. In most areas of mathematics,we expect a superscript to designate an exponent. You might think that wecould reserve superscripts for exponents and use subscripts to designate differentcoordinates or other labels. This approach is sometimes used for so-called flatspaces. However, if we accept Einstein’s Theory of General Relativity, we live ina space that is curved. To reserve superscripts for exponents in the study of curvedspaces is simply too restrictive and inconvenient.

So how can you distinguish a superscript representing an exponent from asuperscript representing some kind of label? If you see a superscript outside of somebracket (usually round), you can be confident that it represents an exponent. Forexample,

.a/2 D aa:

On the other hand, if the meaning is clear from the context, the brackets may beomitted. For example, in the next chapter, I will write c to represent the speed oflight and c2 to represent the square of the speed of light.

I now turn to another issue. Usually, one represents a vector as a linearcombinations of unit row vectors as in (2.1), or a linear combination of unit column

J. Snygg, A New Approach to Differential Geometry using Clifford’s Geometric Algebra,DOI 10.1007/978-0-8176-8283-5 2, © Springer Science+Business Media, LLC 2012

3

4 2 Clifford Algebra in Euclidean 3-Space

vectors. However, as we shall soon see, it is sometimes useful to represent a vectoras a linear combination of square matrices. For example, we could write

x D x1e1 C x2e2 C x3e3; (2.2)

where

e1 D

2

664

0 0 0 1

0 0 1 0

0 1 0 0

1 0 0 0

3

775 ; e2 D

2

664

0 0 1 0

0 0 0 �11 0 0 0

0 �1 0 0

3

775 ; and

e3 D

2

664

1 0 0 0

0 1 0 0

0 0 �1 0

0 0 0 �1:

3

775 (2.3)

At first sight, this may seem to be a pointless variation. However, representinga vector in terms of these square matrices enables us to multiply vectors in a waythat would not otherwise be possible. We should first note that these matrices havesome special algebraic properties. In particular,

.e1/2 D .e2/2 D .e3/2 D I: (2.4)

where I is the identity matrix. Furthermore,

e2e3 C e3e2 D e3e1 C e1e3 D e1e2 C e2e1 D 0: (2.5)

A set of matrices that satisfy (2.4) and (2.5) is said to form the basis for theClifford algebra associated with Euclidean 3-space. There are matrices other thanthose presented in (2.3) that satisfy (2.4) and (2.5). (See Prob. 2.) In the formalismof Clifford algebra, one never deals with the components of any specific matrixrepresentation. We have introduced the matrices of (2.3) only to demonstrate thatthere exist entities that satisfy (2.4) and (2.5).

Now let us consider the product of two vectors. Suppose y D y1e1Cy2e2Cy3e3;then

xy D .x1y1 C x2y2 C x3y3/I C x2y3e2e3 C x3y2e3e2

C x3y1e3e1 C x1y3e1e3 C x1y2e1e2 C x2y1e2e1:

Using the relations of (2.5), we have

xy D .x1y1 C x2y2 C x3y3/I C .x2y3 � x3y2/e2e3

C .x3y1 � x1y3/e3e1 C .x1y2 � x2y1/e1e2: (2.6)

(Note xy ¤ yx.)

2.1 Reflections, Rotations, and Quaternions in E3 5

From (2.6), we can construct formulas for the familiar scalar product hx; yi andthe less familiar wedge product x ^ y. In particular,

hx; yi I D 1

2.xy C yx/ D �

x1y1 C x2y2 C x3y3�

I, and (2.7)

x ^ y D 1

2.xy � yx/ D .x2y3 � x3y2/e2e3

C .x3y1 � x1y3/e3e1 C .x1y2 � x2y1/e1e2. (2.8)

With a slight abuse of notation, we frequently omit the I that appears in (2.7).We note that the coefficients of e2e3, e3e1, and e1e2 that appear in the wedge

product x ^ y are the three components of the cross product x � y.

2.1.2 1-Vectors, 2-Vectors, 3-Vectors, and Clifford Numbers

By considering all possible products of e1, e2 , and e3, one obtains an 8-dimensionalspace spanned by fI; e1; e2; e3; e2e3; e3e1; e1e2; e1e2e3g, where

e2e3 D

2

664

0 0 �1 0

0 0 0 1

1 0 0 0

0 �1 0 0

3

775 ; e3e1 D

2

664

0 0 0 1

0 0 1 0

0 �1 0 0

�1 0 0 0

3

775 ;

e1e2 D

2

664

0 �1 0 0

1 0 0 0

0 0 0 �10 0 1 0

3

775 , and e1e2e3 D

2

664

0 �1 0 0

1 0 0 0

0 0 0 1

0 0 �1 0

3

775 :

One might think that one could obtain higher order products. However, any suchhigher order product will collapse to a scalar multiple of one of the eight matricesalready listed. For example:

e1e2e3e2 D e1e2.e3e2/ D �e1e2.e2e3/ D �e1.e2e2/e3 D �e1e3 D e3e1.

In this fashion, we have obtained an 8-dimensional vector space that is closedunder multiplication. A vector space closed under multiplication is called analgebra. An algebra that arises from a vector space with a scalar product in thesame manner as this example does from E3 is called a Clifford algebra. (We willgive a more formal definition of a Clifford algebra in Chap. 4.)


Fig. 2.1 The vector x’ is theresult of reflecting x withrespect to the planeperpendicular to the unitvector a

a x,a

a

x'

x

I label the matrices e1, e2, and e3 to be Dirac vectors. Any linear combination ofDirac vectors is a 1-vector. A linear combination of e2e3, e3e1, and e1e2 is a 2-vector.In the same vein, a scalar multiple of I is a 0-vector and any scalar multiple of e1e2e3is a 3-vector. A general linear combination of vectors of possibly differing type is aClifford number.

It will be helpful to use an abbreviated notation for products of Dirac vectors. Inparticular, let

e2e3 D e23, e3e1 D e31, e1e2 D e12, and e1e2e3 D e123.

2.1.3 Reflection and Rotation Operators

The algebraic properties of Clifford numbers provide us with a convenient wayof representing reflections and rotations. Suppose a is a vector of unit lengthperpendicular to a plane passing through the origin and x is an arbitrary vector inE3:

(See Fig. 2.1.) In addition, suppose Kx is the vector obtained from x by reflection ofx with respect to the plane corresponding to a. Then

Kx D x � 2 ha; xi a: (2.9)

From (2.7), it is clear that

2 ha; xi a D .ax C xa/ a D axa C x.a/2 D axa C x.

So (2.9) becomesKx D �axa (2.10)

A rotation is the result of two successive reflections (See Fig. 2.2). From Fig. 2.2,it is clear that x is the vector that results from rotating vector x through the angle2 about an axis with the direction of the axial vector a � b. We can rewrite thisrelation in the form:

x D �bKxb D baxab, or

x D R�1xR where R D ab. (2.11)


ψ−θ ψ−θθ

θ

ψ

2ψ

a

b

x''

x'

x

Fig. 2.2 When x is subjectedto two successive reflectionsfirst with respect to a planeperpendicular to a and thenwith respect to a planeperpendicular to b, the resultis a rotation of x about an axisin the direction of a � b. Theangle of rotation is twice theangle between a and b

It is useful to explicitly compute the product ab and interpret the separatecomponents. If

a D a1e1 C a2e2 C a3e3,

and

b D b1e1 C b2e2 C b3e3,

then from (2.7) and (2.8):

R D ab D 1

2.ab C ba/C 1

2.ab � ba/

D I ha;bi C a ^ b.

Since both a and b are vectors of unit length, ha;bi D cos . Furthermore, themagnitude of a � b is sin . Although a ^ b unlike a � b is a 2-vector, a ^ b has thesame three components as a � b. For this reason, we can write

a ^ b D �n1e23 C n2e31 C n3e12

�sin ,

where n1; n2; and n3 are the direction cosines of the axial vector a � b. With thisthought in mind, we have

R D I cos C .n1e23 C n2e31 C n3e12/ sin .


Note! These ideas can be generalized to higher dimensions. For higher dimen-sions the entity a ^ b remains well defined, while a � b becomes meaningless. Inhigher dimensions, you no longer have an axis of rotation; so you must think of therotation as occurring in the 2-dimensional plane spanned by a and b.

We should note that represents 12

the angle of rotation. If � is the actual angleof rotation, we then have

R D I cos�

2C .n1e23 C n2e31 C n3e12/ sin

�

2. (2.12)

To obtain R�1 from R, one can replace � by �� or reverse the order of the Diracvectors. In either case,

R�1 D I cos�

2� .n1e23 C n2e31 C n3e12/ sin

�

2. (2.13)

Returning to (2.11), we see that there appears to be two representations for thesame rotation. In the context of (2.11), R is equivalent to �R. From (2.12), wesee that changing the sign of R is equivalent to replacing � by � C 2� . Indeed,the operator R does not have the expected periodicity of 2� , but it does have aperiodicity of 4� . One’s first reaction is to think that Clifford algebra has introducedan undesirable complication. In the context of (2.11), this may be the case. However,there are circumstance for which this “complication” corresponds to physical reality.We will discuss this point in the next section.

Meanwhile, we note that for k reflections:

Kx D .�1/kakak�1 : : : a1xa1a2 : : : ak D .�1/kT�1xT. (2.14)

2.1.4 Quaternions

Using quaternions, you can represent a rotation operator in a form essentiallyidentical to that which appears in (2.12). What are quaternions? They were invented(discovered?) by William Rowan Hamilton (1805–1865) in 1843. Before thattime, it had been observed that the multiplication of complex numbers could beinterpreted as the multiplication of points in a 2-dimensional plane. This was firstdone by Casper Wessel (1745–1818) in 1797 and then again independently byJean Robert Argand (1768–1822) in 1806 (Kramer 1981, pp. 72–73). In particular,instead of writing:

.aC ib/.c C id / D .ac � bd/C i.ad C bc/, one can write,

.a; b/.c; d / D .ac � bc; ad C bc/.

The question that Hamilton asked himself was, “Could there be a 3-dimensionalversion of this multiplication that would be useful for the study of physics?” Since


his idea was to generalize the notion of complex numbers, he was investigatingtriples of the form: a C ib C jc. You can invent all kinds of multiplication rules,but he was looking for a rule that would be meaningful and useful for the study ofphysics. Starting in 1828, he spent 15 years on this project without success. Finallyon October 16, 1843 (a Monday), he had an eureka experience. He was walkingalong side of the Royal Canal in Dublin with his wife to preside at a Council meetingof the Royal Irish Academy. Then it dawned on him that he should introduce a fourthdimension. In this joyful moment, he carved the formulas for multiplying numbersof the form: aC ibCjcCkd on a stone of the Broome Bridge (or Brougham Bridgeas he called it). ((O’Connor and Robertson: Hamilton) and (Boyer 1968, p. 625)).

Time has obliterated the original carving but in 1958, the Royal Irish Academyerected a plaque commemorating the event:

Here as he walked byon the 16th of October1843

Sir William Rowan Hamiltonin a flash of genius discoveredthe fundamental formula for

quaternion multiplicationi2 D j2 D k2 D ijk D � 1

and cut it in a stone on this bridge.

From the formula that Hamilton carved in stone, it can be shown that

jk D �kj D i, ki D �ik D j, and ij D �ji D k.

(See Prob. 3.)Due to this achievement, William Hamilton is known as the founder of modern

“abstract algebra.”In the theory of quaternions, a rotation operator corresponding to that which

appears in (2.12) is written in the form:

R D I cos�

2� .n1i C n2j C n3k/ sin

�

2. (2.15)

Comparison with (2.12) suggests that we can identify identify i, j, and k,respectively, with �e23, �e31, and �e12. As mentioned above, the binary relationsfor quaternion multiplication are:

.i/2 D .j/2 D .k/2 D �1, (2.16)

jk D �kj D i, (2.17)

ki D �ik D j, and (2.18)

ij D �ji D k. (2.19)


You should check that the same equations hold for the corresponding 2-vectorsassociated with E3. Namely:

.�e23/2 D .�e31/2 D .�e12/2 D �I; (2.20)

.�e31/.�e12/ D �.�e12/.�e31/ D .�e23/; (2.21)

.�e12/.�e23/ D �.�e23/.�e12/ D .�e31/; (2.22)

and .�e23/.�e31/ D �.�e31/.�e23/ D .�e12/: (2.23)

In Hamilton’s formulation, a vector x is represented as x1iCx2j C x3k and therotated vector Kx is computed by the quaternion version of (2.12).

Neither the usual vector formulation nor the Hamilton approach makes a gooddistinction between an ordinary vector and an axial or pseudo-vector.

As we have seen, in the formalism of Clifford algebra, an ordinary vector appearsas a l-vector and a plane of rotation appears as a 2-vector. In three dimensions, a 1-vector and a 2-vector both have three components. In the usual vector formalism,they both appear as 1-vectors. In the quaternion formulation, they both appear as2-vectors.

The distinction between the two entities arises if we consider a reflection. If, forexample, we consider a reflection with respect to the y-z plane, we have

Kx D �e1xe1.

If

x Dx1e1Cx2e2Cx3e3, then

x0 D �x1e1Cx2e2Cx3e3.On the other hand, under the same reflection the 2-vector

X D x1e23Cx2e31Cx3e12 D x1e2e3Cx2e3e1Cx3e1e2becomes

KX D x1.�e1e2e1/.�e1e3e1/C x2.�e1e3e1/.�e1e1e1/C x3.�e1e1e1/.�e1e2e1/ or

KX D x1e23�x2e31�x3e12.This same distinction is carried out in the usual vector formulation but in a

somewhat awkward fashion. Let us consider the cross product x � y. Suppose

x Dx1e1Cx2e2Cx3e3, and

y Dy1e1Cy2e2Cy3e3, then

x � y D.x2y3 � x3y2/e1 C .x3y1 � x1y3/e2 C .x1y2 � x2y1/e3.


How should the cross product transform under a reflection with respect to the y-zplane? If we treat x � y as an ordinary vector, then

.x � y/0D �.x2y3 � x3y2/e1 C .x3y1 � x1y3/e2 C .x1y2 � x2y1/e3.

On the other hand, if we carry out the same reflection on x and y before computingthe cross product, we have

Kx D �x1e1Cx2e2Cx3e3,

Ky D �y1e1Cy2e2Cy3e3, and

Kx � Ky D.x2y3 � x3y2/e1 � .x3y1 � x1y3/e2 � .x1y2 � x2y1/e3.When this second interpretation of the impact of a reflection on x � y is applied,

x � y is said to be an axial or pseudo-vector. In the context of Clifford algebra apseudo-vector is a 2-vector and this awkwardness disappears. Similarly, the entityhx � y, zi ; which is referred to as a pseudo-scalar in the usual vector formulation,becomes a 3-vector in Clifford algebra.

In three dimensions, it is still useful to use the usual cross product, when oneseeks a vector that is perpendicular to a plane spanned by two vectors such as xand y. Thus, we will still use the usual definition:

x � y D.x2y3 � x3y2/e1 C .x3y1 � x1y3/e2 C .x1y2 � x2y1/e3.

However, we will also need the notion of a wedge product that we defined in (2.8).Namely:

x ^ y D1

2.xy � yx/ D .x2y3 � x3y2/e23 C .x3y1 � x1y3/e31 C .x1y2 � x2y1/e12.

In closing this section, we wish to bring to your attention the notion of orthogonaltransformations. An orthogonal transformation is simply a product of reflections.This terminology is chosen when one wishes to focus on the fact that the standardscalar product inEn is preserved. In this chapter, we have restricted ourselves toE3.In this context, it is appropriate that you verify the fact that products of reflectionsdo indeed preserve the scalar product (at least in E3). (See Probs. 6 and 7.)

The product of an even number of reflections (a rotation) is called a properorthogonal transformation, while the product of and odd number of reflections iscalled an improper orthogonal transformation.

Problem 1. From the form of (2.11), it is clear that if the rotation operators R andKR represent two successive rotations, then the combined rotation is represented bythe product R KR. Use this fact and (2.12) to show that a 900 rotation about the y-axisfollowed by a 900 rotation about the x-axis is equivalent to a 1200 rotation about theaxis, which has the direction of the vector .1; 1; 1/.


Problem 2. There are many representations that can be used for e1, e2, and e3. Oneconvenient representation is that using Pauli matrices � 1, � 2, and � 3. That is, wecan let

e1 D �1 D�0 1

1 0

�, e2 D �2 D

�0 �ii 0

�, and e3 D �3 D

�1 0

0 �1�

.

Show that in this representation, (2.4) and (2.5) are satisfied.

Problem 3. If you assume associativity for the multiplication of quaternions, thenusing the equations that appears on Hamilton’s plaque, we have

ijk D � 1 ) .i/2jk D �i ) �jk D �i ) jk D i:

(a) In a similar fashion, show

ki D j and ij D k.

(b) Also show thatkj D �i, ik D �j, and ji D �k.

Problem 4. In the representation introduced in Prob. 2, the quaternions i, j, and kare represented by complex 2 � 2 matrices. In particular,

i D �e23 D �i� 1 D�0 �i

�i 0

�, j D �e31 D �i� 2 D

�0 �11 0

�,

and k D �e12 D �i� 3 D� �i 00 i

�.

In this representation, the rotation operator

R D I cos�

2C .e23n1 C e31n2 C e12n3/ sin

�

2

D"

cos �2

C in3 sin �2.n2 C in1/ sin �

2

�.n2 � in1/ sin �2

cos �2

� in3 sin �2

#

:

Show that in this representation, the matrix representing R is unitary and hasdeterminant equal to 1. (From this result, it is clear that the algebraic propertiesof the double-valued rotation operators for three dimensions can be ascertained bystudying the algebraic properties of 2 � 2 unitary matrices whose determinant is 1.For this reason, the group of double-valued rotation operators is labeled SU.2/: Theletter U indicates “unitary”. The letter S indicates “special”, which in the context ofgroup representation theory means the determinant is 1.)

2.2 The 4� Periodicity of the Rotation Operator 13

Problem 5. Suppose

R D I cos�

2C On sin

�

2, where

On Dn1e23 C n2e31 C n3e12.

(a) Using the fact that

.n1/2 C .n2/2 C .n3/2 D 1, show that

. On/2 D �1:(b) Show that exp

�On. �2/� D R: Hint: represent exp

� On. �2/�

by a Taylor’s series andthen separate the odd and even odd and even powers On.

Problem 6. Suppose Kx D �axa and Ky D �aya, where a is a unit vector. ShowhKx; Kyi D hx; yi. (Remember from (2.7), hx; yi I D 1

2.xy C yx/:)

Problem 7. Suppose Kx D .�1/kakak�1 : : : a1xa1a2 : : : ak and y0 D .�1/kakak�1: : : a1ya1a2 : : : ak . Show hKx; Kyi D hx; yi.

2.2 The 4� Periodicity of the Rotation Operator

From the consequences of the last section, we see that if the vector x.�/ representsthe result of rotating vector x.0/ through an angle � , then we can represent therotation in the form:

x.�/ D R�1.�/x.0/R.�/, where

R.�/ D I cos�

2C On sin

�

2,

On Dn1e23Cn2e31Cn3e12, and

n1, n2, along with n3 are the direction cosines for the axis of rotation.Although x.�/ has a period of 2� , R.�/ has a period of 4�! With the

development of quantum mechanics in the 1920s, it became recognized that a 4�periodicity sometimes occurs in nature. To explain the observed structure of thehydrogen energy spectrum, it was necessary to attribute to the electron a spin of 1

2

and a periodicity of 4� . Later, it became recognized that some objects larger thanelectrons also have a 4� periodicity (Bolker 1973). A demonstration of this fact hasbeen put forward by Edgar Riefin (1979).

For an object to display a 4� periodicity, it is necessary that it be in some senseattached to its surroundings.


Fig. 2.3 A book with a 4� periodicity

To illustrate this, you may wish to carry out a demonstration. First, hold a glass ofwater in the palm of your hand. The hand holding the glass may be left or right butit is important that your hand be under the glass with palm up. Then maintain a firmgrip on the glass and rotate it 3600 without moving your feet or spilling any water.

2.3 *The Point Groups for the Regular Polyhedrons 15

When you have completed this maneuver, you will find yourself in an awkwardposition with the glass slightly above your head and your elbow pointed upward.Clearly, the relationship of the glass to you is quite different from what it was in itsinitial position. However, if you continue the rotation, you may be surprised to findthat your arm will unwind itself and the glass will return to its initial position withits initial relationship to you. Thus, the glass attached to your arm does not have a2� periodicity but it does have a 4� periodicity.

This demonstration is shown in Fig. 2.3 where a book is used in place of a glassof water.

2.3 *The Point Groups for the Regular Polyhedrons

One aspect of geometry, which attracts a lot of attention in physics, is symmetrygroups. The symmetry of a body can be characterized by the set of transformationsthat maintain distances between points and bring the body into its original spaceof occupation. Quite reasonably, these are called symmetry transformations. Forinfinite bodies (for example an infinite crystal lattice), the set of symmetry transfor-mations may contain translations.

But for finite bodies, symmetry transformations are restricted to rotations andproducts of rotations and reflections. For this reason, Clifford algebra is a good toolto attack the mathematics of symmetry for finite bodies.

Before getting very deep into this topic, it is useful to prove a theorem byElie Cartan (1938, pp. 13–17; 1966, pp. 10–12). His theorem states that in ann-dimensional space (real or complex), a transformation consisting of any finitenumber of reflections can also be obtained by a number of reflections that does notexceed n.

In this text, we only need the real 3-dimensional version and that is the onlyversion we will prove.

Theorem 8. Suppose Kx D .�1/kak ak�1 : : : a1xa1a2 : : : ak . That is we have atransformation consisting of k reflections. Then this same transformation (in E3)can be achieved by three or fewer reflections.

Proof. Case 1. The number of reflections k is even. If we multiply an even numberof 1-vectors, we get a linear combination of the 0-vector I and the three 2-vectorse23, e31, and e21. That is

a1a2 : : : ak D I˛ C e23ˇ1 C e31ˇ2 C e12ˇ3.

(This already looks like a rotation operator!) The operator akak�1 : : : a1 is essen-tially the same as a1a2 : : : ak except for the fact that the underlying Dirac vectorsare in reverse order. Thus,

akak�1 : : : a1 D I˛ C e32ˇ1 C e13ˇ2 C e21ˇ3 D I˛ � e23ˇ1 � e31ˇ2 � e12ˇ3.


Since .a1/2 D .a2/2 D : : : D .ak/2 D I; .akak�1 : : : a1/ .a1a2 : : : ak/ D I, and

I D �I˛ � e23ˇ1 � e31ˇ2 � e12ˇ3

� �I˛ C e23ˇ1 C e31ˇ2 C e12ˇ3

�

D I�.˛/2 C .ˇ1/2 C .ˇ2/2 C .ˇ3/2

�:

Since .˛/2 C .ˇ1/2 C .ˇ2/2 C .ˇ3/2 D 1, there exists an angle such that

cos D ˛ and sin Dp.ˇ1/2 C .ˇ2/2 C .ˇ3/2.

Furthermore, if at least one of the ˇk’s is not zero, we can define the directioncosines for the axis of rotation by

nk D ˇk=p.ˇ1/2 C .ˇ2/2 C .ˇ3/2 D ˇk= sin for k D 1; 2; and 3.

(Note! this definition guarantees that .n1/2 C .n2/2 C .n3/2 D 1.) We now haveshown:

a1a2 : : : ak D I cos C .n1e23 C n2e31 C n3e12/ sin .

If the sin D 0; a1a2 : : : ak D ˙I: Otherwise, we have a nontrivial rotationoperator. From Fig. 2.2, it is clear that this rotation operator can be replaced bya product of two reflections.

Case 2. The number of reflections k is odd.In this case, we can multiply out the first k-1 reflections to get a rotation operator

and we then have:

a1a2 : : : ak D �I cos C .n1e23 C n2e31 C n3e12/ sin

�ak

D �I cos C .n1e23 C n2e31 C n3e12/ sin

�.k1e1 C k2e2 C k3e3/:

If sin D 0 or k1n1 C k2n2 C k3n3 D 0, our product a1a2 : : : ak reduces toa 1-vector. Otherwise after factoring the rotation into two reflections, we have theproduct of three reflections. utNow we are in a position to have a reasonably intelligent discussion of symmetrygroups. Generally, the set of multiple reflections that bring a particular finite bodyinto its original position in space is called a point group for two reasons. One is dueto the fact that at least one point remains fixed under all symmetry transformationsassociated with a particular body. The second is due to the fact that the set of thesymmetry transformations identified with a particular body forms a mathematicalstructure known as a group.

Definition 9. A group is a set of elements with a binary operation ı having thefollowing properties:

(1) Closure: g1 2 G, g2 2 G ) g1 ı g2 2 G:(2) Associativity: .g1 ı g2/ ı g3 D g1 ı .g2 ı g3/:


Tetrahedron Cube Octahedron

IcosahedronDodecahedron

Fig. 2.4 The five regular polyhedrons

(3) Identity element: 9 an element e 2 G such that 8 g 2 G; e ı g D g ı e D g.(4) Inverse: 8 g 2 G, 9 g�1 2 G such that g ı g�1 D g�1 ı g D e.

Examples of groups include the integers under addition, the positive rationalnumbers under multiplication, and nonsingular n�n matrices under matrix multi-plication.

We will only give a short description of a few point groups – in particular the fivepoint groups associated with the five regular polyhedrons. (See Fig. 2.4.) For eachof the polyhedrons, we have a finite symmetry group. One way to verify we have agroup is to run through the check list in the definition above.

The elements of a symmetry group for a finite solid are finite products ofreflections. It is clear that the multiplication of two finite products results in a finiteproduct, which preserves the original position of the relevant solid. Thus, the set ofsymmetry transformations satisfy the property of closure.

The identity element corresponds to the transformation that does nothing orrotates the solid some integral multiple of 3600:

To obtain the inverse of a product of reflections, one simply constructs theproduct of the same reflections in the reverse order.

To show that the symmetry groups for the regular polyhedrons have only afinite number of members, let us consider the example of the cube. (See Fig. 2.5.)Applying Cartan’s theorem, we know that an even number of reflections (a properorthogonal transformation) can be reduced to either the identity element or arotation. The possible symmetry rotations are not difficult to count.


A

B

CFig. 2.5 Some symmetryaxes of rotation for the cube

Perhaps, the most obvious symmetry rotations are those that correspond to thefourfold axes that pass through the centers of opposite faces. Not counting the 3600

identity rotation, we have symmetry rotations of 900, 1800, and 2700. Since thereare three such axes, this gives us 3 � 3 D 9 elements.

We also have some twofold axes that pass through the midpoints of oppositeedges. Since there are 12 edges, there are six such axes and corresponding to eachof these axes is a symmetry rotation of 1800. This accounts for six more elementsin the group. Then there are four threefold axes that pass through opposite vertices.This adds another eight members to the group.

Finally, there is the identity transformation. Thus, the total number of properorthogonal members for the point group associated with the cube is 9 C 6 C 8 C1 D 24. (Because any product of reflections has two representations in the Cliffordformalism (˙/; there are 48 Clifford numbers in the Clifford version of the properorthogonal group for the cube.)

To obtain the number of improper orthogonal transformations by simply countingthem is difficult because some members of this set are not simple reflections butproducts of three reflections. To complete our counting problem, we wish to applythe following theorem:

Theorem 10. For a finite point group, the number of improper orthogonal trans-formations (products of an odd number of reflections) is equal to the number ofproper transformations (products of an even number of reflections). Note! Forthose familiar with group theory, what is proven below is that the set of improperorthogonal transformations is a coset of the subgroup of proper orthogonaltransformations.

Proof. To establish the truth of this theorem, we choose a unit vector a correspond-ing to a simple reflection in the group and then show that any improper orthogonaltransformation can be represented uniquely (aside from the sign ambiguity) in theform Ra where R is a rotation or ˙ the identity element I:


Consider a product of an odd number of reflections a1a2 : : : ak . If k is odd,we can multiply out the first k-1 reflections to get a rotation operator R. So wehave

a1a2 : : : ak D Rak.

If ak D a, we are incredibly lucky. Otherwise,

Rak D Rak.a/2 D R.aka/a D R KRa D Ra, where

R D R KR. Thus, we have

a1a2 : : : ak D Ra.

To show that this representation is unique aside from the sign ambiguity, suppose

Ra D ˙ KRa. Multiply both sides by a to get

R.a/2 D ˙ KR.a/2 or R D ˙ KR: ut

Applying this theorem to the cube, we see that the point group for the cube has48 members (96 for the double valued Clifford version).

Using the terminology of group theory, we say the order of the point group forthe cube is 48.

To get the orders for the point groups of the other polyhedrons, the chief problemis counting the edges and vertices. For example, the dodecahedron is constructed byassembling 12 regular pentagons. Before assembly, the 12 pentagons have a total of12 � 5 D 60 edges. When assembled, one edge from one pentagon and one edgefrom a second pentagon align to become a single edge of the dodecahedron. Thus,the dodecahedron has 60=2 D 30 edges, which correspond to 30=2 D 15 twofoldaxes. Similarly, the 60 vertices of the 12 pentagons become 60=3 D 20 vertices forthe dodecahedron. In turn, this corresponds to ten threefold axes.

For four of the five regular polyhedrons, the axes of symmetry pass through pairsof faces, pairs of edges, or pairs of vertices. The one exception is the tetrahedron. Forthe tetrahedron, the twofold axes do indeed correspond to pairs of edges. Howeverfor the threefold axes, the situation is different. For the tetrahedron, each threefoldaxis passes through one vertex and one face.

When you determine the orders of the point groups (See Prob. 12.), you willsee that the order of the point group for the cube is identical to the order of thepoint group for the octahedron. This raises the possibility that the two groups areisomorphic. Two groups are said to be isomorphic if one can set up a one-to-onecorrespondence between the groups is such a way that if x in one group correspondsto Kx in the second group and y corresponds to Ky then x ı y corresponds to Kx ı Ky.For the cube and the octahedron, this is plausible because the numbers of fourfold,threefold, and twofold axes match up in the two groups. Nonetheless, it would


a b

Fig. 2.6 (a) A cube aligned with a skeleton frame of an octahedron. (b) An icosahedron alignedwith a skeleton frame of a dodecahedron

be very difficult to determine an isomorphic correspondence without resorting togeometry. However using geometry, it becomes a trivial exercise to establish theisomorphism. One merely matches the vertices of one with the face centers of theother. In Fig. 2.6a, we have aligned a cube with the skeleton frame of an octahedronis such a way that the symmetry axes of rotation for the two polyhedrons coincide.Thus we see that a proper symmetry transformation for one of the polyhedrons isa proper symmetry transformation for the other. The two point groups also containthe same improper symmetry transformations. (See Prob. 15.) Thus, the two pointgroups are isomorphic.

In Fig. 2.6b, we have aligned an icosahedron with the skeleton frame of adodecahedron with similar consequences.

One can also demonstrate geometrically that the point group for the tetrahedronis a subgroup of the point groups for the other polyhedrons so that any symmetrytransformation of the tetrahedron is also a symmetry transformation of the otherpolyhedrons.

One can imbed a tetrahedron inside a cube so that the threefold axes for thetwo polyhedrons coincide. (See Fig. 2.7a.) The twofold axes of the tetrahedron donot coincide with the twofold axes of the cube. However, the twofold axes of thetetrahedron do coincide with the fourfold axes of the cube. Thus, it becomes clearthat any proper orthogonal transformation in the point group for the tetrahedronbelongs to the point group for the cube. It can also be said that any impropertransformation belonging to the point group for the tetrahedron is also an impropertransformation belonging to the point group for the cube. (See Prob. 16.) Thus, it isclear that the point group for the tetrahedron is a subgroup of the point group for thecube.

It is more difficult to visualize but the point group for the tetrahedron is also asubgroup of the dodecahedron (or icosahedron). (See Fig. 2.7b.)


x

y

za b

Fig. 2.7 (a) A tetrahedron aligned with the skeleton frame of a cube. (b) A tetrahedron alignedwith the skeleton frame of a dodecahedron

A

B

C

D

E

Fig. 2.8 A cube aligned withthe skeleton frame of adodecahedron.

It is also enlightening to examine Fig. 2.8. You may not be convinced thatconnecting some of the vertices of the dodecahedron as shown in Fig. 2.8 results inthe edges of a cube. However, it should be clear that the direction of line segment ABis perpendicular to the direction of line segment DE. Furthermore, line segment DEis parallel to line segment BC. Thus, the edges of our suspect cube do indeed meetat right angles at each vertex. By studying the alignment of the various symmetryaxes of rotation in Fig. 2.8, we reach the conclusion that the intersection of the pointgroup for the cube (or octahedron) and the point group for the dodecahedron (oricosahedron) is the point group for the tetrahedron.


Problem 11. Prove that there are no more than five regular polyhedrons. Hint:What is the maximum number of equilateral triangles that can share a single vertex?

Problem 12. Determine the orders of the point groups for the tetrahedron, octahe-dron, dodecahedron, and icosahedron. Are your results consistent with Figs. 2.6aand 2.6b?

Problem 13. How does the result of Prob. 1 relate to the point group for the cube?What is the consequence of two successive 900 rotations about two non-alignedfourfold axes?

Problem 14. In view of Fig. 2.7a, the three twofold axes of the tetrahedron canbe aligned with the x, y, and z axes. Suppose we designate a 1800 rotation aboutthe x-axis by Rx D ˙e23. Suppose we also define Ry and Rz in a similar manner.Complete the following table:

ı I Rx Ry Rz

IRx Rx

Ry

Rz

You will find that the 1800 rotations commute, although the Clifford representa-tions do not.

Problem 15. Consider Fig. 2.6a.

(a) Draw the figure with the cube and octahedron aligned with the x, y, and z axes.(b) Describe a plane of reflection that is common to both the cube and octahedron.(c) It has already been pointed out that if the cube and the octahedron are aligned

as in Fig. 2.6a, the proper orthogonal symmetry transformations for the twopoint groups are identical. Use your result in part b) to show that the impropersymmetry transformations for the two point groups are identical.

(d) Explain why the improper symmetry transformations for the icosahedron arethe same as the improper symmetry transformations for the dodecahedron.

Problem 16. (a) Prove that any improper orthogonal symmetry transformation forthe tetrahedron is also an improper orthogonal symmetry transformation for thecube. (If you get stuck, review the approach used in the proof of Theorem 10.)

(b) Explain why any improper orthogonal symmetry for the tetrahedron is alsoan improper orthogonal symmetry transformation for the dodecahedron (oricosahedron).

Problem 17. If a tetrahedron is aligned with the x, y, and z axes as shown inFig. 2.7a, then the rotations about the threefold axis shown are

˙�

I cos 600 C sin 600�1p3

e23 C 1p3

e31 � 1p3

e12

�D ˙1

2ŒI C e23 C e31 � e12/�

2.4 *Elie Cartan 1869–1951 23

and

˙�

I cos 1200 C sin 1200�1p3

e23 C 1p3

e31 � 1p3

e12

�D �1

2ŒI � e23 � e31 C e12/�

D ˙1

2ŒI � e23 � e31 C e12/� :

(a) List all of the rotations for both the twofold and threefold axes. (Don’t computethem all – after computing a few, you should see patterns.)

(b) Write down the Clifford representation of a reflection and use this to constructa list of the improper orthogonal symmetry for the tetrahedron.

(c) In the list constructed in part b), which are simple reflections and which cannotbe achieved by fewer than three reflections?

Problem 18. Euler’s FormulaIn 1750, Leonard Euler made the conjecture that for any convex polyhedron,

F � E C V D 2, where F equals the number of faces, E equals the number ofedges, and V equals the number of vertices (James 2002, p. 5). Determine whetherthis formula is valid for the five regular polyhedrons. Suppose you slice off a cornerof a cube. Does the resulting solid satisfy Euler’s formula?

2.4 *Elie Cartan 1869–1951

The way mathematicians deal with differential geometry was significantly alteredby the work of Elie Cartan. In 1993, the American Mathematical Society publisheda 301-page translation from Russian of a summary of his work. This short biographyis extracted from that source.

The authors of that summary are two Russian mathematicians: M.A. Akivis andB.A. Rosenfeld (1993). Elie Cartan’s contributions to mathematics are so deep andbroad that these two accomplished geometers felt compelled to include a virtualapology in their preface: “Of course the authors are only able to describe in detailCartan’s results connected with those branches of geometry in which the authors areexperts.” (Akivis and Rosenfeld 1993, p. xi).

Elie Joseph Cartan was born on April 9, 1869 in Dolomieu, a small village insoutheastern France of less than 2,000 people. At the time of his birth, no one wouldhave predicted that Elie Cartan would become a world renowned mathematician.His father was a blacksmith. His older sister, Jeanne-Marie, became a dressmaker,and his younger brother, Leon, would eventually join the family business as anotherblacksmith.

Elie seemed destined for a similar career in rural France until a fateful visit toElie’s elementary school by the up and coming politician, Antonin Dubost (1844–1921). This event would change Elie’s direction in life.


When Elie’s teachers described their very remarkable student to Dubost, Dubostencouraged the young Cartan to compete for a scholarship at a more competitivelycee. Antonin Dubost eventually became the Minister of Justice under one adminis-tration and later became President of the French Senate for what was essentially thelast 14 years of his life. Throughout his life, Antonin Dubost maintained a fatherlyinterest in Cartan’s career.

To help Elie obtain the desired scholarship, one of his teachers, M. Dupuis,supervised his preparation for the required exam. Cartan scored well on the exam,received the scholarship, and left home at the age of 10.

At the age of 17, Cartan decided to become a mathematician and enrolled atl’Ecole Normale Superieure in Paris. During the next three years, Cartan not onlyattended lectures at l’Ecole Normale Superieure but also at the Sorbonne. In thisway, he became exposed to many outstanding mathematicians including HenriPoincare. After graduation, he was drafted into the French army for one year. Hethen returned to Paris and received his doctorate at the Sorbonne two years later in1894 while attracting the attention of prominent mathematicians including SophusLie at Leipzig University in Germany.

Early in his career, Cartan developed aspects of Lie groups and Lie algebrasthat could be applied to differential geometry. Later, his work on differential formsled him to develop methods that are now commonly used to deal with differentialequations. In 1910, Cartan began to perfect the method of moving frames to dealwith problems in differential geometry (Cartan 1910a, 1910b). (You will encounterthis method in later chapters of this book.)

In 1915, when Cartan was 46, he was again drafted into the French army soonafter World War I broke out. However, he was not sent to the front. Instead, he wasassigned to a hospital set up in the building of l’Ecole Normale Superieure. Thissituation allowed him to continue his mathematical research during the war years.

During these same war years, Einstein living in Berlin, discovered that a slightvariation of Riemannian geometry was necessary to express his general theory ofrelativity. After the war, Einstein and others sought out mathematical structures thatcould be used to construct a unified field theory. With this motivation, Cartan turnedhis attention to extracting properties of more general geometric spaces that mightbe useful. (His correspondence with Einstein was edited by Robert Debever andpublished by Princeton University Press in 1979 under the title Elie Cartan andAlbert Einstein: Letters on Absolute Parallelism, 1929–1932.)

To summarize, Cartan was prolific. Akivis and Rosenfeld attribute over 200publications to Cartan, and this includes several books that have been republishedin recent years.

Cartan was also successful as a family person. In 1903, he married Marie-Louise Bianconi (1880–1950) and soon became the father of three sons: Henri(1904–2008), Jean (1906–1932), and Louis (1909–1943). Later Elie and Marie-Louise had a daughter Helene (1917–1952). His first son, Henri, became a worldrenowned mathematician in his own right. (Henri Cartan died on August 13, 2008at the age of 104!) His second son, Jean, seemed headed for a promising career

2.5 *Suggested Reading 25

as a music composer but he died of tuberculosis at the age of 25. The third son,Louis, was a talented physicist, but during World War II, he was arrested by Vichygovernment police for his activities in the French resistance. He was then turnedover to the Germans who held him in captivity for 15 months before executinghim by decapitation. The daughter Helene taught mathematics at several lycees andauthored several math papers before she died at the age of 34.

During most of his adult life, Elie Cartan made his home in Paris or withincommuting distance of Paris. He had spent much of his boyhood away from hishometown but he always maintained his ties there. He encouraged his younger sisterAnna to pursue a career in math education. She taught at several secondary schoolsfor girls and authored two textbooks, which were reprinted many times.

In 1909, Cartan built a vacation home in Dolomieu and sometimes he couldbe seen at the family blacksmith shop helping his father and brother to blow theblacksmith bellows.

Cartan’s sister Anna and daughter Helene were not the only women to receiveCartan’s encouragement to study mathematics. After he retired from his professorialposition at the Sorbonne in 1940, he devoted the last years of his life in his 70s toteaching mathematics at the Ecole Normale Superieure for girls.

After a long illness, he died in Paris on May 6, 1951.

2.5 *Suggested Reading

Milton Hamermesh 1962. Group Theory. Reading, Massachusetts, U.S.A: Addison-Wesley Publishing Company, Inc. Also reprint edition 1990. New York. DoverPublications, Inc.

The second chapter is devoted to the point groups.Leo Dorst, Chris Doran, and Joan Lasenby (Editors) 2002. Applications of

Geometric Algebra in Computer Science and Engineering. Boston: Birkhauser.Chapter I entitled “Point Groups and Space Groups in Geometric Algebra”by DavidHestenes is devoted to the application of Geometric Algebra (Clifford Algebra) tothe classification of symmetry groups.

D.M.Y. Sommerville 1958. An Introduction to the Geometry of N Dimensions.New York: Dover Publications, Inc.

This book includes a discussion of regular polyhedrons in higher dimensions.

http://www.springer.com/978-0-8176-8282-8

Clifford Algebra in Euclidean 3-Space

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